Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces by GE-Evolution Operator Method

Zhaoqiang Ge

doi:10.3390/math11030743

journal article Open Access Feb 02, 2023

Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces by GE-Evolution Operator Method

Zhaoqiang Ge

Mathematics Vol. 11 No. 3 pp. 743 · MDPI AG

View at Publisher Save 10.3390/math11030743

Abstract

Controllability is a basic problem in the study of stochastic generalized systems. Compared with ordinary stochastic systems, the structure of stochastic singular systems is more complex, and it is necessary to study the controllability of stochastic generalized systems in the context of different solutions. In this paper, the controllability of semilinear stochastic generalized systems was investigated by using a GE-evolution operator for integral and impulsive solutions in Hilbert spaces. Some sufficient and necessary conditions were obtained. Firstly, the existence and uniqueness of the integral solution of semilinear stochastic generalized systems were discussed using the GE-evolution operator theory and Banach fixed point theorem. The existence and uniqueness theorem of the integral solution was obtained. Secondly, the approximate controllability of semilinear stochastic generalized systems was studied in the case of the integral solution. Thirdly, the existence and uniqueness of the impulsive solution of semilinear stochastic generalized systems were considered, and some sufficient conditions were provided. Fourthly, the approximate controllability of semilinear stochastic generalized systems was studied for the impulsive solution. At last, the exact controllability of linear stochastic systems was studied in the case of the impulsive solution, with some necessary and sufficient conditions given. The obtained results have important theoretical and practical value for the study of controllability of semilinear stochastic generalized systems.

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References

49

[1]

Mahmudov "On controllability of semilinear stochastic systems in Hilbert spaces" IMA J. Math. Control Inf. (2002) 10.1093/imamci/19.4.363

[2]

Mahmudov "Controllability of semilinear stochastic systems in Hilbert spaces" J. Math. Anal. Appl. (2003) 10.1016/s0022-247x(03)00592-4

[3]

Mahmudov "Approximate controllability of semilinear deterministic and stochastic evolution equation in abstract spaces" SIAM J. Control Optim. (2003) 10.1137/s0363012901391688

[4]

Mahmudov "Controllability of non-linear stochastic systems" Int. J. Control (2003) 10.1080/0020717031000065648

[5]

Mahmudov "Controllability of semilinear stochastic systems" Int. J. Control (2005) 10.1080/00207170500207180

[6]

Dauer "Approximate controllability of backward stochastic evolution equations in Hilbert spaces" J. Math. Anal. Appl. (2006) 10.1016/j.jmaa.2005.09.089

[7]

Sakthivel "On controllability of nonlinear stochastic systems" Rep. Math. Phys. (2006) 10.1016/s0034-4877(06)80963-8

[8]

Fu "Controllability of non-densely neutral functional systems in abstract space" Chin. Ann. Math. Ser. B (2007) 10.1007/s11401-005-0028-9

[9]

Sakthivel "Controllability of nonlinear impulsive stochastic systems" Int. J. Control (2009) 10.1080/00207170802291429

[10]

Muthukumar "Approximate controllability for semilinear retarded stochastic systems in Hilbert spaces" IMA J. Math. Control Inf. (2009) 10.1093/imamci/dnp004

[11]

Sakthivel "Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay" Taiwan J. Math. (2010) 10.11650/twjm/1500406016

[12]

Nutan "Approximate controllability of non-densely defined semilinear delayed control systems" Nonlinear Stud. (2011)

[13]

Ning "Approximate controllability of nonlinear stochastic partial differential systems with infinite delay" Adv. Differ. Equ. (2015) 10.1186/s13662-015-0434-6

[14]

Shukla "Complete controllability of semilinear stochastic systems with delay" Rend. Circ. Mat. Palermo (2015) 10.1007/s12215-015-0191-0

[15]

Mokkedem "Approximate controllability for a semilinear stochastic evolution systems with infinite delay in Lp space" Appl. Math. Optim. (2017) 10.1007/s00245-016-9332-x

[16]

Rajivganthi "Existence and approximate controllability of stochastic semilinear reaction diffusion systems" J. Danam. Control (2017)

[17]

Arora "Approximate controllability of non-densely defined semilinear control system with nonlocal conditions" Nonlinear Dyn. Syst. Theory (2017)

[18]

Anguraj, A., and Ramkumar, K. (2018). Approximate controllability of semilinear stochastic integrodifferential system with nonlocal conditions. Fractal Fract., 2. 10.3390/fractalfract2040029

[19]

Shukla "Controllability of semilinear stochastic control system with finite delay" IMA J. Math. Control Inf. (2018)

[20]

Mokkedem "Approximate controllability for a retarted semilinear stochastic evolution system" IMA J. Math. Control Inf. (2019) 10.1093/imamci/dnx045

[21]

Anguraju "Approximate controllability of a semilinear impulsive stochastic system with nonlocal conditions and Poisson jumps" Adv. Differ. Equ. (2020) 10.1186/s13662-019-2461-1

[22]

Xu "Approximate controllability for semilinear second-order stochastic evolution systems with infinite delay" Int. J. Control (2020) 10.1080/00207179.2018.1518597

[23]

Huang "Approximate controllability of semilinear stochastic integro-differential equations with infinite delay" IMA J. Math. Control Inf. (2020) 10.1093/imamci/dnz040

[24]

Balch "Statistical null-controllability of stochastic nonlinear parabolic equations" Stocha. PDE Anal. Comp. (2021)

[25]

Bashirov, A.E. (2021). C-controllability of stochastic semilinear systems. Math. Method Appl. Sci. 10.22541/au.162840278.84661841/v1

[26]

Lu, Q., and Zhang, X. (2021). Mathematical Control Theory for Stochastic Partial Differential Equations, Springer. 10.1007/978-3-030-82331-3

[27]

Liaskos "Linear stochastic degenerate Sobolev equations and applications" Int. J. Control (2015) 10.1080/00207179.2015.1048482

[28]

Liaskos "Stochastic degenerate Sobolev equations: Well posedness and exact controllability" Math. Meth. Appl. Sci. (2018) 10.1002/mma.4077

[29]

Dzektser "Generalization of the equation of motion of ground waters with a free surface" Engl. Trans. Sov. Phys. Dokl. (1972)

[30]

Ge "Approximate controllability and approximate observability of singular distributed parameter systems" IEEE Trans. Autom. Control (2020) 10.1109/tac.2019.2920215

[31]

Exact Controllability and Exact Observability of Descriptor Infinite Dimensional Systems

Zhaoqiang Ge

IEEE/CAA Journal of Automatica Sinica 2021 10.1109/jas.2020.1003411

[32]

Melnikova "Abstract stochastic equations I, classical and distribution solutions" J. Math. Sci. Funct. Anal. (2002)

[33]

Vlasenko "Stochastic impulse control of parabolic systems of Sobolev type" Differ. Equ. (2011) 10.1134/s0012266111100132

[34]

Vlasenko "Optimal control of a class of random distributed Sobolev type systems with aftereffect" J. Autom. Inf. Sci. (2013) 10.1615/jautomatinfscien.v45.i9.60

[35]

Palanisamy "Approximate boundary controllability of Sobolev type stochastic differential systems" J. Egypt. Math. Soc. (2014) 10.1016/j.joems.2013.07.005

[36]

Generalized stochastic evolution equations

Kenneth L. Kuttler, Ji Li

Journal of Differential Equations 2014 10.1016/j.jde.2014.04.017

[37]

Approximate Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces

Zhaoqiang Ge

Mathematics 10.3390/math10173050

[38]

Ge "Solvability of a time-varying singular distributed parameter systems in Banach space" Sci. China Inf. Sci. (2013)

[39]

Da Prato, G., and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions, Cambridge University Press. 10.1017/cbo9780511666223

[40]

Ge "GE-evolution operator method for controllability of time varying stochastic descriptor systems in Hilbert spaces" IMA J. Math. Control Inf. (2022) 10.1093/imamci/dnab038

[41]

Curtain, R., and Zwart, H.J. (1995). An Introduction to Infinite Dimensional Linear Systems Theory, Springer. 10.1007/978-1-4612-4224-6

[42]

Liu "Exact controllability for stochastic Schrodinger equations" J. Differ. Equ. (2013)

[43]

Liu "Exact controllability for stochastic transport equations" SIAM J. Control Optim. (2014) 10.1137/130910373

[44]

Ge, Z.Q. (2021). Review of the latest progress in controllability of stochastic linear systems and stochastic GE-evolution operator. Mathematics, 9. 10.3390/math9243240

[45]

Hu, S.G., Huang, C.M., and Wu, F.K. (2008). Stochastic Equations, Science Press.

[46]

Oksenda, B. (1998). Stochastic Differential Equations: An Introduction with Application, Springer.

[47]

Mao, X. (1997). Stochastic Differential Equation and Their Application, Horwood Publishing.

[48]

Zhang "Some remarks on stability of stochastic singular systems with state depend noise" Automatica (2015) 10.1016/j.automatica.2014.10.044

[49]

Bonaccorsi "Stochastic variation of constants formula for infinite dimensional equation" Stoch. Anal. Appl. (1999) 10.1080/07362999908809616

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Details

Published: Feb 02, 2023
Vol/Issue: 11(3)
Pages: 743
License: View

Authors

Z

Zhaoqiang Ge

School of Mathematics and Statistics, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an 710049, China

Funding

National Natural Science Foundation of China Award: 12126401

Cite This Article

Zhaoqiang Ge (2023). Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces by GE-Evolution Operator Method. Mathematics, 11(3), 743. https://doi.org/10.3390/math11030743

Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces by GE-Evolution Operator Method

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